Inequalities:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Univ. Press
1967
|
| Ausgabe: | 2. ed., reprint. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 324 S. |
Internformat
MARC
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| 245 | 1 | 0 | |a Inequalities |c G. H. Hardy ; J. E. Littlewood ; G. Pólya |
| 250 | |a 2. ed., reprint. | ||
| 264 | 1 | |a Cambridge |b Univ. Press |c 1967 | |
| 300 | |a XII, 324 S. | ||
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Datensatz im Suchindex
| _version_ | 1804117259785863168 |
|---|---|
| adam_text | TABLE OF CONTENTS
Chapter I. INTRODUCTION
Theorems Pages
1.1. Finite, infinite, and integral inequalities . . 1 2
1.2. Notations 2
1.3. Positive inequalities 2—3
1.4. Homogeneous inequalities 3 4
1.5. The axiomatic basis of algebraic inequalities 4—5
1.6. Comparable functions 5 6
1.7. Selection of proofs 6 8
1.8. Selection of subjects 8 11
Chapter II. ELEMENTARY MEAN VALUES
2.1. Ordinary means 12 13
2.2. Weighted means 13 14
2.3. Limiting oases of SBl,(a) 1 5 14 15
2.4. Cauchy s inequality 6 8 16
2.5. The theorem of the arithmetic and geometric
means 9 16 18
2.6. Other proofs of the theorem of the means . 18 21
2.7. Holder s inequality and its extensions . . 10 11 21 24
2.8. Holder s inequality and its extensions (coni.). 12 15 24 26
2.9. General properties of the means SM ) • . 16 17 26 28
2.10. Thesum3 Sr(a) 18 23 28 30
2.11. Minkowski s inequality 24 26 30 32
2.12. A companion to Minkowski s inequality . . 27 28 32
2.13. Illustrations and applications of the funda¬
mental inequalities 29 36 32 37
2.14. Inductive proofs of the fundamental in¬
equalities 37 40 37 39
2.15. Elementary inequalities connected with
Theorem 37 41 42 39 42
2.16. Elementary proof of Theorem 3 42 43
2.17. Tchebychef s inequality 43 44 43 44
2.18. Muirhead s theorem 45 44 45
viii TABLE OE1 CONTENTS
Theorems Pages
2.19. Proof of Muirhead s theorem 46 48
2.20. An alternative theorem 46 47 49
2.21. Further theorems on symmetrical means . . 48 50 49 51
2.22. The elementary symmetric functions of n
positive numbers 51 55 51 55
2.23. A note on definite forms 55 57
2.24. A theorem concerning strictly positive forms 56 57 57 60
Miscellaneous theorems and examples . . . 58 81 60 64
Chaptbb III. MEAN VALUES WITH AN ARBITRARY
FUNCTION AND THE THEORY OF
CONVEX FUNCTIONS
3.1. Definitions 82 65 66
3.2. Equivalent means 83 66 68
3.3. A characteristic property of the means SB, . 84 68 69
3.4. Comparability 85 69 70
3.5. Convex functions 70 71
3.6. Continuous convex functions 86 87 71 72
3.7. An alternative definition 88 89 73 74
3.8. Equality in the fundamental inequalities . . 90 91 74 75
3.9. Restatements and extensions of Theorem 85 92 93 75 76
3.10. Twice differentiate convex functions . . . 94 95 76 77
3.11. Applications of the properties of twice differ
entiable convex functions 96 97 77 78
3.12. Convex functions of several variables . . . 98 99 78 81
3.13. Generalisations of Holder s inequality . . . 100 101 81 83
3.14. Some theorems concerning monotonic func¬
tions 102 104 83 84
3.15. Sums with an arbitrary function: generalisa¬
tions of Jensen s inequality 105 84 85
3.16. Generalisations of Minkowski s inequality . 106 85 88
3.17. Comparison of sets 107 110 88 91
3.18. Further general properties of convex functions 111 91 94
3.19. Further properties of continuous convex
functions 112 94 96
3.20. Discontinuous convex functions 96
Miscellaneous theorems and examples . . . 113 139 97 101
TABLE OF CONTENTS iX
Chapter IV. VARIOUS APPLICATIONS
OF THE CALCULUS
Theorems Pages
4.1. Introduction 102
4.2. Applications of the mean value theorem . . 140 143 102 104
4.3. Further applications of elementary differential
calculus 144 148 104 106
4.4. Maxima and minima of functions of one
variable 149 150 106 107
4.5. Use of Taylor s series 151 107
4.6. Applications of the theory of maxima and
minima of functions of several variables . 108 110
4.7. Comparison of series and integrals 152 155 110 111
4.8. An inequality of W. H. Young 156 160 111 113
Chapteb V. INFINITE SERIES
5.1. Introduction 114 116
5.2. The means 9R, 116 118
5.3. The generalisation of Theorems 3 and 9 . . 118 119
5.4. Holder s inequality and its extensions . . . 161 162 119 121
5.5. The means SK, (cont.) 163 121 122
5.6. The sums Sr 164 122 123
5.7. Minkowski s inequality 165—166 123
5.8. Tchebychef s inequality 123
5.9. A summary 167 123 124
Miscellaneous theorems and examples , . . 168—180 124—125
Chapter VI. INTEGRALS
6.1. Preliminary remarks on Lebesgue integrals . 126 128
6.2. Remarks on null sets and null functions . . 128 129
6.3. Further remarks concerning integration . . 129 131
6.4. Remarks on methods of proof 131 132
6.5. Further remarks on method: the inequality
of Schwarz 181 182 132 134
6.6. Definition of the means SJt, (/) when r =t=0 . 183 134 136
6.7. The geometric mean of a function .... 184 186 136 138
6.8. Further properties of the geometric mean . 187 139
6.9. Holder s inequality for integrals .... 188 191 139 143
X TABLE OF CONTENTS
Theorems Pages
6.10. General properties of the means !Dt,.(/) . . 192 194 143 144
6.11. General properties of the means 3Rr (/) (cont.) 195 144 145
6.12. Convexity of log SK, 196 197 145 146
6.13. Minkowski s inequality for integrals . . . 198 203 146 150
6.14. Mean values depending on an arbitrary
function 204 206 150 152
6.15. The definition of the Stieltjes integral . . . 152 154
6.16. Special cases of the Stieltjes integral . . . 154 155
6.17. Extensions of earlier theorems 155 156
6.18. The means 2ft,. (/; l ) 207 214 156 157
6.19. Distribution functions 157 158
6.20. Characterisation of mean values 215 158 159
6.21. Remarks on the characteristic properties . . 160 161
6.22. Completion of the proof of Theorem 215 . . 161 163
Miscellaneous theorems and examples . . . 216 252 163 171
Chapter VII. SOME APPLICATIONS OF THE
CALCULUS OF VARIATIONS
7.1. Some general remarks ........ 172 174
7.2. Object of the present chapter 174 175
7.3. Example of an inequality corresponding to
an unattained extremum 253 254 175 176
7.4. First proof of Theorem 254 176 178
7.5. Second proof of Theorem 254 255 178 182
7.6. Further examples illustrative of variational
methods 256 182 184
7.7. Further examples: Wirtinger s inequality . 257 258 184 187
7.8. An example involving second derivatives. . 259 260 187 193
7.9. A simpler problem 261 193
Miscellaneous theorems and examples . . . 262 272 193 195
Chapteb VIII. SOME THEOREMS CONCERNING
BILINEAR AND MULTILINEAR FORMS
8.1. Introduction 196
8.2. An inequality for multilinear forms with
positive variables and coefficients . . . 273 275 196 198
8.3. A theorem of W. H. Young 276 277 198 200
8.4. Generalisations and analogues 278 284 200 202
TABLE OF CONTENTS xi
Theorems Pages
8.5. Applications to Fourier series 202 203
8.6. The convexity theorem for positive multi¬
linear forms 285 203 204
8.7. General bilinear forms 286 288 204 206
8.8. Definition of a bounded bilinear form . . . 206 208
8.9. Some properties of bounded forms in [p, q] . 289 290 208 210
8.10. The Faltung of two forms in [p, p ] ... 291 210 211
8.11. Some special theorems on forms in [2, 2] . . 292 293 211 212
8.12. Application to Hilbert s forms 294 212 214
8.13. The convexity theorem for bilinear forms with
complex variables and coefficients . . . 295 214 216
8.14. Further properties of a maximal set (x, y) . 216 217
8.15. Proof of Theorem 295 217 219
8.16. Applications of the theorem of M. Riesz . . 296 297 219 220
8.17. Applications to Fourier series 220 221
Miscellaneous theorems and examples . . 298 314 222 225
Chapter IX. HILBEET S INEQUALITY AND ITS
ANALOGUES AND EXTENSIONS
9.1. Hubert s double series theorem 315 317 226 227
9.2. A general class of bilinear forms .... 318 227 229
9.3. The corresponding theorem for integrals . . 319 229 230
9.4. Extensions of Theorems 318 and 319 ... 320 322 231 232
9.5. Best possible constants: proof of Theorem 317 232 234
9.6. Further remarks on Hilbert s theorems . . 323 234 236
9.7. Applications of Hilbert s theorems .... 324 325 236 239
9.8. Hardy s inequality 326 327 239 243
9.9. Further integral inequalities 328 330 243 246
9.10. Further theorems concerning series . . . 331 332 246 247
9.11. Deduction of theorems on series from theorems
on integrals 333 247 249
9.12. Carleman s inequality 334 335 249 250
9.13. Theorems with 0 p l 336 338 250 253
9.14. A theorem with two parameters p and q . . 339 340 253 254
Miscellaneous theorems and examples . , . 341 367 254 209
Xii TABLE OF CONTENTS
Chapteb X. REARRANGEMENTS
Theorems Pages
10.1. Rearrangements of finite sets of variables . 260 261
10.2. A theorem concerning the rearrangements of
two sets 368 369 261 262
10.3. A second proof of Theorem 368 262 264
10.4. Restatement of Theorem 368 370 264 265
10.5. Theorems concerning the rearrangements of
three sets 371 373 265 266
10.6. Reduction of Theorem 373 to a special case . 266 268
10.7. Completion of the proof 268 270
10.8. Another proof of Theorem 371 270 272
10.9. Rearrangements of any number of sets . . 374 376 272 274
10.10. A further theorem on the rearrangement of
any number of sets 377 274 276
10.11. Applications 276
10.12. The rearrangement of a function .... 276 278
10.13. On the rearrangement of two functions . . 378 278 279
10.14. On the rearrangement of three functions . 379 279 281
10.15. Completion of the proof of Theorem 379 . 281 284
10.16. An alternative proof 285 287
10.17. Applications 380 383 288 291
10.18. Another theorem concerning the rearrange¬
ment of a function in decreasing order . . 384 385 291 292
10.19. Proof of Theorem 384 292 295
Miscellaneous theorems and examples . . 386—405 295 299
Appendix I. On strictly positive forms . . . 406 407 300
Appendix II. Thorin s proof and extension of
Theorem 295 408 305
Appendix III. On Hubert s inequality. . . . 308
Bibliography 310
|
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| indexdate | 2024-07-09T15:51:34Z |
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| language | English |
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| physical | XII, 324 S. |
| publishDate | 1967 |
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| spelling | Hardy, Godfrey H. 1877-1947 Verfasser (DE-588)118720376 aut Inequalities G. H. Hardy ; J. E. Littlewood ; G. Pólya 2. ed., reprint. Cambridge Univ. Press 1967 XII, 324 S. txt rdacontent n rdamedia nc rdacarrier Calcul Inégalités (Mathématiques) Ongelijkheden gtt Inequalities (Mathematics) Ungleichung (DE-588)4139098-2 gnd rswk-swf Ungleichung (DE-588)4139098-2 s DE-604 Littlewood, John Edensor 1885-1977 Verfasser (DE-588)118780182 aut Pólya, George 1887-1985 Verfasser (DE-588)118825321 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001858543&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hardy, Godfrey H. 1877-1947 Littlewood, John Edensor 1885-1977 Pólya, George 1887-1985 Inequalities Calcul Inégalités (Mathématiques) Ongelijkheden gtt Inequalities (Mathematics) Ungleichung (DE-588)4139098-2 gnd |
| subject_GND | (DE-588)4139098-2 |
| title | Inequalities |
| title_auth | Inequalities |
| title_exact_search | Inequalities |
| title_full | Inequalities G. H. Hardy ; J. E. Littlewood ; G. Pólya |
| title_fullStr | Inequalities G. H. Hardy ; J. E. Littlewood ; G. Pólya |
| title_full_unstemmed | Inequalities G. H. Hardy ; J. E. Littlewood ; G. Pólya |
| title_short | Inequalities |
| title_sort | inequalities |
| topic | Calcul Inégalités (Mathématiques) Ongelijkheden gtt Inequalities (Mathematics) Ungleichung (DE-588)4139098-2 gnd |
| topic_facet | Calcul Inégalités (Mathématiques) Ongelijkheden Inequalities (Mathematics) Ungleichung |
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